Inverse approach to solutions of the Dirac equation for space-time dependent fields
IInverse approach to solutions of the Dirac equation for space-time dependent fields
Johannes Oertel and Ralf Sch¨utzhold ∗ Fakult¨at f¨ur Physik, Universit¨at Duisburg-Essen, Lotharstrasse 1, 47057 Duisburg, Germany (Dated: September 20, 2018)Exact solutions of the Dirac equation in external electromagnetic background fields are very help-ful for understanding non-perturbative phenomena in quantum electrodynamics (QED). However,for the limited set of known solutions, the field often depends on one coordinate only, which could bethe time t , a spatial coordinate such as x or r , or a light-cone coordinate such as ct − x . By swappingthe roles of known and unknown quantities in the Dirac equation, we are able to generate families ofsolutions of the Dirac equation in the presence of genuinely space-time dependent electromagneticfields in 1 + 1 and 2 + 1 dimensions. PACS numbers: 03.65.Pm, 11.15.Tk, 12.20.Ds
I. INTRODUCTION
Quantum electrodynamics (QED) as the theory ofcharged particles interacting with electromagnetic fieldsis well understood in the context of standard perturba-tion theory and can describe several intriguing phenom-ena of nature. However, QED contains other fascinat-ing effects that cannot be explained using perturbativemethods. Such non-perturbative effects can arise whenthe electromagnetic field is so strong that it cannot betreated as a perturbation. In order to understand thesephenomena, it is often useful to study the behaviour ofexact solutions in those external background fields.Unfortunately, although the Dirac equation was firstformulated more than eighty years ago [1], the set ofknown exact solutions is still quite limited (see, e.g., [2]for a review). Apart from the Coulomb field ∝ /r [3, 4], exact solutions are known for a constant electricfield and a Sauter profile in space ∝ / cosh ( kx ) or time ∝ / cosh ( ωt ), for example. The latter are relevant forthe non-perturbative Sauter-Schwinger effect [5–7] corre-sponding to electron-positron pair creation from vacuumvia tunnelling. In contrast to electron-positron pair cre-ation in the perturbative (multi-photon) regime whichhas been observed at SLAC [8], this non-perturbativeprediction of quantum field theory has not been conclu-sively experimentally verified yet. However, there areseveral experimental initiatives which might be able toeventually reach the ultra-strong field regime necessaryfor observing this striking effect [9].Furthermore, exact solutions are known for a constantmagnetic field (relativistic Landau levels, see [10, 11])and plane waves, where the fields depend on one of thelight-cone coordinates such as ct − x (Volkov solutions,see e.g. [12–15]). These (transverse) fields do not inducepair creation from vacuum.Nevertheless, in all these cases, the fields depend onone coordinate only (such as r , x , t , or ct − x ). As aresult of this high degree of symmetry, the set of partial ∗ [email protected] differential equations can be reduced to an ordinary dif-ferential equation, which greatly simplifies the analysis.An analogous limitation applies to our theoretical un-derstanding of the Sauter-Schwinger effect. Even thoughthere are many results for fields which depend on onecoordinate only, we are just beginning to understand theimpact of the interplay between spatial and temporal de-pendencies, see, e.g., [16–21].In the following, we develop a method which allows usto obtain solutions of the Dirac equation for genuinelyspace-time dependent fields. To this end, we pursue adifferent approach by assuming that we already know asolution to the Dirac equation. We then calculate thevector potential A µ corresponding to the given solutionfrom the Dirac equation. This is feasible as the Diracequation does not contain any derivatives of the vectorpotential. More generally speaking, we write down a so-lution to a partial differential equation and then try tofind a physical problem associated with the solution – aconcept also well known in the field of fluid dynamics,see, for example, [22]. II. LIGHT CONE COORDINATES
Let us start with the most simple and yet non-trivialcase – the Dirac equation in 1+1 dimensions. For thefollowing derivation, it is convenient to transform to lightcone coordinates x ± defined as (¯ h = c = 1) x + = t + x √ , x − = t − x √ . (1)Since perhaps not all readers will be familiar with theform of the subsequent expressions in light cone coordi-nates, let us insert a brief reminder. The Jacobian matrixof the coordinate transformation between Cartesian andlight cone coordinates J µν = ∂ ( x + , x − ) ∂ ( t, x ) = 1 √ (cid:18) − (cid:19) , (2) a r X i v : . [ h e p - t h ] M a r yields the transformation laws for tensors such as thepartial derivatives ∂ µ (cid:48) = ( J − ) νµ ∂ ν = 1 √ (cid:18) ∂ t + ∂ x ∂ t − ∂ x (cid:19) = (cid:18) ∂ x + ∂ x − (cid:19) = .. (cid:18) ∂ + ∂ − (cid:19) . (3)In 1+1 dimensions, the electromagnetic field strengthtensor contains only one independent component, theelectric field E ( t, x ) F µν = ∂ µ A ν − ∂ ν A µ = (cid:18) E − E (cid:19) , (4)which thus reads in light cone coordinates F (cid:48) µν = ( J − ) λµ ( J − ) ρν F λρ = (cid:18) − EE (cid:19) . (5)Transforming the Cartesian Minkowski metric tensor η µν = diag(+1 , −
1) to light cone coordinates as well gives g µν = (cid:18) (cid:19) . (6)A possible choice of light cone gamma matrices satisfyingthe Clifford algebra’s anti-commutation relation { γ µ , γ ν } = 2 g µν , (7)therefore is γ + = (cid:18) √
20 0 (cid:19) , γ − = (cid:18) √ (cid:19) . (8)Note that in 1+1 and 2+1 dimensions, the Clifford alge-bra can be satisfied with 2 × III. INVERSE APPROACH
The Dirac equation, minimally coupled to the electro-magnetic potential A µ via the charge q (cid:0) iγ µ (cid:2) ∂ µ + iqA µ (cid:3) − m (cid:1) ψ = 0 , (9)assumes the following simple form in terms of the lightcone gamma matrices (8) (cid:18) − m i √ (cid:2) ∂ + + iqA + (cid:3) i √ (cid:2) ∂ − + iqA − (cid:3) − m (cid:19) (cid:18) ψ ψ (cid:19) = 0 . (10)Traditionally, the Dirac equation is treated as a partialdifferential equation. A solution ψ for a specific potential A µ is typically calculated by reducing the Dirac equationto an ordinary differential equation. In our approach, weassume that we know a specific spinor ψ = ( ψ , ψ ) T which is a solution to the Dirac equation and calculatethe corresponding potential. Thus, we solve (10) for thecomponents of A µ qA + = i ∂ + ψ ψ − m √ ψ ψ ,qA − = i ∂ − ψ ψ − m √ ψ ψ . (11) For arbitrary ψ , these expressions are not necessarily real.Therefore, we require the imaginary parts of qA + and qA − to vanish, giving two conditions which we use toeliminate two real degrees of freedom of the spinor ψ .Using the polar representation for the spinor components ψ k = r k e iϕ k , these conditions can be written as r ∂ + r − m √ r r sin ( ϕ − ϕ ) = 0 ,r ∂ − r + m √ r r sin ( ϕ − ϕ ) = 0 . (12)Adding the two equations gives ∂ − r = − ∂ + r , (13a) ∂ − r = m √ r sin ( ϕ − ϕ ) . (13b)The first equation (13a) can be solved for r by integrat-ing with respect to x + r = (cid:115) c ( x − ) − (cid:90) ∂ − r d x + , (14)where c ( x − ) is an integration constant that may still de-pend on x − . The remaining equation (13b) determinesthe phase difference ϕ − ϕ ϕ − ϕ = arcsin (cid:32) √ m ∂ − r r (cid:33) , (15)where we could also use other branches of the arcsin-function such as ∆ ϕ = ϕ − ϕ → π − ∆ ϕ , leading to dif-ferent solutions in general – see the remark after Eq. (17).Using the abbreviation s = (cid:115) c − (cid:90) ∂ − r d x + − m ( ∂ − r ) , (16)and Eqs. (14) and (15), we can calculate the form of thespinor ψ ψ = (cid:18) ψ ψ (cid:19) = e iϕ (cid:18) r ± s + i √ m ∂ − r (cid:19) , (17)where we have set r = r and ϕ = ϕ . Note that wefind two different solutions with ± s corresponding to thedifferent branches of the arcsin or square-root functionsin Eqs. (15) and (16), respectively.Local gauge invariance allows us to eliminate the phase e iϕ by applying a gauge transformation ψ (cid:55)→ ψ (cid:48) = e − iϕ ψ ,which adds a term ∂ µ ϕ to qA µ . The components of A µ using the spinor given in (17) finally are qA + = ∓ m √ rs ∓ √ m ∂ + ∂ − rs ,qA − = ∓ m √ sr . (18)These are obviously real as long as r and s are real, too.The electric field corresponding to this potential accord-ing to (5) is E = ∂ − A + − ∂ + A − . (19)In summary, by choosing a real generating function r ( x + , x − ) and a real supplementary boundary value func-tion c ( x − ), we can generate arbitrary space-time depen-dent solutions ψ ( x + , x − ) of the Dirac equation in thepresence of an electromagnetic background A µ , whichcan also depend on space and time.Obviously, the associated electromagnetic fieldstrength tensor F µν in Eq. (5) automatically satisfies thehomogeneous Maxwell equations as it has been derivedfrom a vector potential A µ . If we demand that it alsoobeys the inhomogeneous Maxwell equations ( µ = 1) ∂ ν F µν = j µ , (20)we have to specify the sources j µ accordingly. In 1 + 1dimensions we find ρ = − ∂ x E = 1 √ (cid:0) ∂ − E − ∂ + E (cid:1) ,j = ∂ t E = 1 √ (cid:0) ∂ − E + ∂ + E (cid:1) , (21)where ρ is the charge density and j is the current density.For non-trivial field profiles E ( t, x ), they will be non-zero in general. However, this is no surprise because theonly vacuum solution of the Maxwell equations in 1 + 1dimensions is a constant electric field E = const. IV. SOLUTIONS
In order to illustrate the approach presented in theprevious section, let us discuss some exemplary solutionsthat can be found using this method, starting with themost simple ones. The expressions for the spinor and thepotential components are significantly simplified if r isindependent of x − . A. Plane waves
Choosing r and s = ±√ c to be constant, ψ = (cid:18) rs (cid:19) = const , (22)leads to a constant electromagnetic vector potential qA + = − m √ rs = const ,qA − = − m √ sr = const . (23) Thus, a gauge transformation ψ (cid:55)→ ψ (cid:48) = e − ip µ x µ ψ with p µ .. = (cid:18) p + p − (cid:19) = m √ (cid:18) r/ss/r (cid:19) (24)can be used to set the potential components to zero andreveals that these solutions are plane wave solutions tothe free Dirac equation of either positive or negative en-ergy. Transforming the light-cone momenta p ± back tothe usual Cartesian representation p = ( p + + p − ) / √ p = m ( r/s + s/r ) / r and s are positive (or bothnegative) corresponds to a positive energy whereas dif-ferent signs of r and s yield a negative energy. B. Single pulses
In this subsection, we find solutions for arbitrary lightcone fields E ( x + ) and E ( x − ), i.e., pulses moving alongthe light lines. Such solutions were found before usingtraditional methods as well [14, 15]. x + -dependent pulse Let us assume that the function r depends on x + onlywhile s = ±√ c = const ψ = (cid:18) r ( x + ) s (cid:19) . (25)In this case, neither the spinor nor the vector potentialdepends on x − which simplifies the expression for theelectric field qE = q ∂ − A + (cid:124) (cid:123)(cid:122) (cid:125) =0 − q ∂ + A − = m √ s ∂ + r ( x + ) . (26)This is a first-order ordinary differential equation for r ( x + ) which can be integrated easily r ( x + ) = r in (cid:34) √ m r in s q (cid:90) x + −∞ E (˜ x + ) d ˜ x + (cid:35) − , (27)with r in = r ( x + → −∞ ). Comparison with Sec. IV A re-veals that the pre-factor √ r in / ( ms ) in front of the above x + -integral over qE is just the inverse initial momen-tum 1 /p in − . As already discussed in [14], the term in thesquare bracket in Eq. (27) vanishes and thus r divergeswhen this x + -integral over qE becomes large enough tocompensate p in − . Note that the light cone dispersion rela-tion p + p − = m / p + must diverge when p − vanishes and vice versa.Now, let us recall that the phenomenon of electron-positron pair creation (such as in the Sauter-Schwingereffect) can be described by the situation where an ini-tial solution with positive energy transforms into a finalsolution which contains contributions with negative en-ergies (or vice versa). Assuming that r becomes constantinitially and finally, we find that pair creation can onlyoccur if r ( x + ) changes its sign somewhere, i.e., if r ( x + )vanishes or diverges at some point. If r ( x + ) crosses zero,the electric field (26) diverges – whereas a diverging r ( x + )precisely corresponds to the case discussed above, see also[14]. Thus, we find that we cannot describe particle cre-ation in this case without introducing some singularity(see the Appendix). x − -dependent pulse In a similar way, we can derive solutions for electricfields only depending on x − by setting r = const andletting s ( x − ) = ± (cid:112) c ( x − ) depend on x − ψ = (cid:18) rs ( x − ) (cid:19) . (28)Thus, the electric field can be calculated as follows qE = q ∂ − A + − q ∂ + A − (cid:124) (cid:123)(cid:122) (cid:125) =0 = − m √ r ∂ − s ( x − ) , (29)which is again a first-order ordinary differential equationfor s ( x − ). The solution is given by s ( x − ) = s in (cid:34) − √ m s in r q (cid:90) x − −∞ E (˜ x − ) d ˜ x − (cid:35) − , (30)with s in = s ( x − → −∞ ). In complete analogy, the samearguments as for an x + -dependent pulse apply in thiscase. C. Two pulses
As a non-trivial extension of these two cases, we cancombine the previous two solutions into a single spinor ψ = (cid:18) r ( x + ) s ( x − ) (cid:19) , (31)where the two components are given by r ( x + ) = r in (cid:34) √ m r in s in q (cid:90) x + −∞ E + (˜ x + ) d ˜ x + (cid:35) − ,s ( x − ) = s in (cid:34) − √ m s in r in q (cid:90) x − −∞ E − (˜ x − ) d ˜ x − (cid:35) − . (32)We may calculate the electric field using (18) and (19) E ( x + , x − ) = r ( x + ) r in E − ( x − ) + s ( x − ) s in E + ( x + ) . (33) FIG. 1. Plot of r ( x + , x − ) as given in (34) with r in = 1, ξ = 0 .
2, and γ = 1 . /m . Initially, we have E ( x + , x − ) = E − ( x − ) + E + ( x + ) whichcorresponds to two independent pulses approaching eachother from different directions. When these two pulsesmeet, however, this is no longer true – which shows thatthe mapping from ψ (i.e., r and s ) to A µ is not linear. Forlate times, these pulses propagate again independently,but with modified amplitudes in general. D. Emerging pulses
Another solution where the corresponding electric fieldconsists of two pulses can be generated by setting r ( x + , x − ) = r in + ξ e − γx + + e − γx − . (34)For non-vanishing ξ and γ >
0, the chosen r ( x + , x − ) willbe constant almost everywhere except in the vicinity ofthe forward light cone (see figure 1).In this case, the expression for s according to (16) isnot as simple as before because r is not independent of x − . Nevertheless, s can be calculated analytically, al-though the resulting expressions for s and the electricfield qE are quite lengthy. Thus, we will only give a plotof the resulting electric field which shows the two pulsesemerging from the origin and moving along the forwardlight lines (see figure 2). V. EXTENSION TO 2+1 DIMENSIONS
The approach presented here can be extended to 2 + 1dimensional space-times as well. We use the Cartesiancoordinate y in addition to the light cone coordinates x + FIG. 2. Plot of the electric field qE corresponding to thesolution generated by r ( x + , x − ) given in (34) with r in = 1, ξ = 0 .
2, and γ = 1 . /m . and x − . Thus, the metric tensor becomes g µν = − . (35)In order to complete our set of gamma matrices from (8),we choose the third γ -matrix according to γ = iσ z = (cid:18) i − i (cid:19) . (36)Thus the Dirac equation in 2 + 1 dimensions is given by (cid:18) − m − (cid:2) ∂ y + iqA y (cid:3) i √ (cid:2) ∂ + + iqA + (cid:3) i √ (cid:2) ∂ − + iqA − (cid:3) − m + (cid:2) ∂ y + iqA y (cid:3)(cid:19) (cid:18) ψ ψ (cid:19) = 0 . (37)In complete analogy to section III, we solve the Diracequation for qA + and qA − and reduce the spinor’s num-ber of degrees of freedom by requiring the imaginaryparts of the electromagnetic potential’s components tovanish. After some calculation, we are able to write thespinor and the electromagnetic potential in terms of threereal functions r ( x + , x − , y ), r ( x + , x − , y ) and c ( x + , x − ).Explicitly, a spinor of the form ψ = (cid:18) r s − iu (cid:19) , (38)with s = ± (cid:113) r − u (39)and u = 1 √ r (cid:20) c ( x + , x − ) + (cid:90) (cid:0) ∂ − r + ∂ + r (cid:1) d y (cid:21) (40) is a solution of the Dirac equation with the potentialcomponents qA + = − m √ r s − √ ∂ y r s + ∂ + us ,qA − = − m √ r r s + 1 √ r ∂ y r r s − u ∂ − r r s ,qA y = − m us − u ∂ y r r s + 1 √ ∂ + r r s . (41)In contrast to 1+1 dimensions, the electromagnetic fieldstrength tensor contains three independent components,for example the two electric fields E x,y in x and y di-rection plus the perpendicular magnetic field B z . Thesecomponents of the electromagnetic field can be calculatedas follows E x = ∂ − A + − ∂ + A − ,E y = 1 √ (cid:0) ∂ − A y − ∂ y A − + ∂ + A y − ∂ y A + (cid:1) ,B z = 1 √ (cid:0) ∂ − A y − ∂ y A − − ∂ + A y + ∂ y A + (cid:1) . (42)We see that these expressions simplify significantly if r and r are independent of y . In that case, the electromag-netic field does only depend on the light cone coordinatesas before and similar solutions as in the 1+1 dimensionalcase can be found, e.g. one and two wavefronts. In fact,the solutions given in section IV are solutions to the 2+1dimensional Dirac equation as well but can be extendedto also include a transverse electric and magnetic fieldcomponent.To verify that our method reproduces known solutions,we insert the lowest Landau level solution ψ = N exp (cid:32) − qB (cid:20) x − k y qB (cid:21) (cid:33) (cid:18) (cid:19) (43)into our formalism, i.e. we set r ( x + , x − ) = N exp (cid:32) − qB (cid:20) x + − x − √ − k y qB (cid:21) (cid:33) ,r ( x + , x − ) = r ( x + , x − ) , c = 0 , (44)where N is a normalization constant. Calculating thepotential components gives qA + = qA − = − m √ , qA y = − qB x + − x − √ k y , (45)so that the electromagnetic field is E x = E y = 0 , B z = B, (46)which is the expected result. VI. CONCLUSIONS & OUTLOOK
We have developed an inverse approach for generatingfamilies of exact solutions of the Dirac equation in thepresence of space-time dependent electromagnetic fieldsin 1+1 and 2+1 dimensions. Somewhat similar to op-timal control theory, we start with a suitable ansatz forthe spinor ψ and then derive the appropriate backgroundfield A µ which supports this solution. In 1+1 dimensions,we may choose a real generating function r ( x + , x − ) anda suitable real supplementary boundary value function c ( x − ) such that the radicand in Eq. (14) stays positive.In 2+1 dimensions, we may choose two real generatingfunctions r ( x + , x − , y ) and r ( x + , x − , y ) as well as onereal boundary value function c ( x + , x − ).The solutions generated in this way may depend onspace and time in a complicated manner – a situationwhich is quite difficult to treat with traditional meth-ods. As one possible application, our method could beused to solve steering problems such as: given an initialwave-packet ψ in , which electromagnetic field A µ inducesan evolution to a prescribed final wave-packet ψ out ? Asanother application, these exact solutions could be usedas touchstones for already existing exact or approximatenon-perturbative derivation techniques (e.g. the world-line instanton method [23]) or as starting point for newapproximative methods, such as WKB [24] or lineariza-tion around a given background solution (see the Ap-pendix).The structure of the Dirac equation suggests that thisgeneral strategy can also be applied to 3+1 dimensions,where both the potential A µ and the Dirac bi-spinor ψ have four components. Thus, for a given ψ , we getfour equations for the four components A µ , which can besolved (except in singular cases). However, the four con-straints (cid:61) ( A µ ) = 0 assume a form which is far more com-plicated than in 1+1 and 2+1 dimensions. This rendersthe identification of real generating functions which cor-respond to the remaining degrees of freedom rather cum-bersome. The analysis could be simplified by restrictingthe space-time dependence to 1+1 and 2+1 dimensions,which should be the subject of further investigations. ACKNOWLEDGMENTS
R.S. acknowledges support by DFG (SFB-TR12) andwould like to express special thanks to the Perimeter In-stitute for Theoretical Physics and the Mainz Institutefor Theoretical Physics (MITP) for hospitality and sup-port.
Appendix A: Perturbed solution
To find solutions for electric fields that create electron-positron pairs (see, e.g., [25–28]), we use the ansatz r = α + β sin( mγ ) , (A1)where the Bogoliubov coefficients α and β as well as theeikonal function γ are slowly varying functions of thelight cone coordinates. (We consider 1+1 dimensions forsimplicity.) The main idea here is that α is an exactsolution and β is used to slowly turn on an oscillatingperturbation. The value of β then is related to the paircreation rate.However, the calculation of s and qE is rather com-plicated for arbitrary functions α , β , and γ because s depends nonlinearly on r . Hence, as the perturbationshould be small, we calculate the electric field only up tolinear order in βE = E ( α ) + E ( β ) + O ( β ) , (A2)where qE ( α ) is the unperturbed force of order β and qE ( β ) is the first-order perturbation of order β . Apartfrom this linearization, we assume that the mass m rep-resents the largest energy scale in the problem and thuswe employ a large- m expansion on top of the approxima-tion in Eq. (A2). Expanding qE ( β ) into powers of m andkeeping only the highest-order term gives qE ( β ) = √ β cos( mγ ) s α ∂ + γ (cid:34) m ( ∂ + γ ) ( ∂ − γ ) − (cid:18) m √ αs α (cid:124) (cid:123)(cid:122) (cid:125) = − qA ( α )+ ∂ − γ + m √ s α α (cid:124) (cid:123)(cid:122) (cid:125) = − qA ( α ) − ∂ + γ (cid:19) (cid:35) + O ( m ) , (A3)with the abbreviation s α = (cid:115) c − (cid:90) ∂ − α d x + . (A4)( qA ( α )+ and qA ( α ) − are the leading-order contributions tothe vector potential.) Since α , β , and γ are supposed tobe slowly varying, the leading contribution (A3) wouldbe rapidly oscillating due to the pre-factor cos( mγ ) un-less the phase function γ has a stationary point (see be-low). Of course, such a rapidly oscillating force with a fre-quency of order m could well create pairs, but this processwould be typically in the perturbative (multi-photon)regime. Here, we are interested in non-perturbative phe-nomena such as the Sauter-Schwinger effect and thus wedemand that these rapidly oscillating contributions areabsent – at least to leading order. Thus, we require theterm of order m in qE ( β ) to vanish. This is the case if S = mγ solves the eikonal equation m (cid:16) ∂ + S + qA ( α )+ (cid:17) (cid:16) ∂ − S + qA ( α ) − (cid:17) . (A5)Therefore, this condition can be used to fix γ for a given α . Then, the leading order of qE ( β ) is of order m qE ( β ) = m √ s α sin( mγ ) (cid:40) ∂ + β ) (cid:20) ∂ − γ∂ + γ + (cid:18) αs α ∂ − γ∂ + γ (cid:19) (cid:21) + 2( ∂ − β ) (cid:20) (cid:16) s α α (cid:17) ∂ + γ∂ − γ (cid:21) + β (cid:20) ∂ + αα (cid:18) ∂ − γ∂ + γ + (cid:18) αs α ∂ − γ∂ + γ (cid:19) + 2 (cid:16) s α α (cid:17) (cid:19) + ∂ − αα (cid:18) (cid:16) s α α (cid:17) ∂ + γ∂ − γ + (cid:18) αs α (cid:19) (cid:18) ∂ − γ ) − ∂ − γ∂ + γ (cid:19) (cid:19) + ∂ + ( ∂ − γ ) + (cid:18) αs α (cid:19) ∂ − γ∂ + γ ∂ + (cid:18) ∂ − γ∂ + γ (cid:19) (cid:21)(cid:41) + O ( m ) , (A6)where we have used the eikonal equation (A5) to sim-plify some expressions. If we require this rapidly oscil-lating term to vanish as well, we get a linear first-orderpartial differential equation for β . However, this linearequation does not have any source term. Therefore, asolution where β vanishes initially will not generate anypairs unless the coefficients of ∂ + β and ∂ − β vanish at some point. 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